Which NN Arch? - Given input vec X -> Output vector Y made of "not binary", floating point values
This is a regression problem instead of a classification problem. For both, you can use a feedforward neural network. These are the typical ones you've probably seen where each neuron in a layer is connected to each neuron in the next.

In a classification problem, you want an output like you mentioned above, a one-hot vector corresponding to the correct class. The way you get this vector is by applying a soft-max activation to the outputs of your last layers. If you don't apply this activation, your output vector might be something like [-2.5, 0.6, 10.2, 5.1]. You can see that this looks a lot like the vector you want. So, you can probably use a feedforward architecture (with non-linear activations on the hidden layers if there are non-linearities in your data) with a final layer of 4 neurons. Then, when you train, that should (hopefully) converge to the correct data outputs. If it doesn't, you will need to mess with the architecture/get more training data/lots of other levers to play with.

This link goes to a tutorial for regression using Keras. Keras is a nice package written to take powerful libraries like TensorFlow and Theano and give them a simple interface. TensorFlow can be quite complex (I've never used Theano so I don't know) when you're starting and trying to understand the syntax and the ML concepts. Keras helps take away the confusing syntax.